Albert Einstein Center for Fundamental Physics Institute ... · String Dynamics in Yang-Mills...

String Dynamics in Yang-Mills Theory

Uwe-Jens Wiese

Albert Einstein Center for Fundamental PhysicsInstitute for Theoretical Physics, Bern University

Workshop on “Strongly Interacting Field Theories”Jena, October 1, 2010

Collaboration: Ferdinando Gliozzi (INFN Torino)Michele Pepe (INFN Milano)

Uwe-Jens Wiese

Outline

Yang-Mills Theory

Systematic Low-Energy Effective String Theory

Luscher-Weisz Multi-Level Simulation Technique

String Width at Zero and at Finite Temperature

String Breaking and String Decay

Exceptional Confinement in G (2) Yang-Mills Theory

Conclusions

Outline

Yang-Mills Theory

Conclusions

Yang-Mills theory

Gµ(x) = igG aµ(x)T a, a ∈ 1, 2, . . . , nG,

Gµν(x) = ∂µGν(x)− ∂νGµ(x) + [Gµ(x),Gν(x)],

S [Gµ] =

∫d4x

4g2Tr[GµνGµν ]

Main sequence gauge groups

group SU(N) Sp(N) SO(2N + 1) SO(4N) SO(4N + 2)

nG N2 − 1 2N2 + N 2N2 + N 8N2 − 2N 8N2 + 6N + 1rank N − 1 N N 2N 2N + 1

center Z(N) Z(2) Z(2) Z(2)× Z(2) Z(4)

Exceptional gauge groups

group G (2) F (4) E (6) E (7) E (8)

nG 14 52 78 133 248rank 2 4 6 7 8

center 1 1 Z(3) Z(2) 1

Yang-Mills theory

Gµ(x) = igG aµ(x)T a, a ∈ 1, 2, . . . , nG,

S [Gµ] =

∫d4x

4g2Tr[GµνGµν ]

center Z(N) Z(2) Z(2) Z(2)× Z(2) Z(4)

group G (2) F (4) E (6) E (7) E (8)

nG 14 52 78 133 248rank 2 4 6 7 8

center 1 1 Z(3) Z(2) 1

Yang-Mills theory

Gµ(x) = igG aµ(x)T a, a ∈ 1, 2, . . . , nG,

S [Gµ] =

∫d4x

4g2Tr[GµνGµν ]

center Z(N) Z(2) Z(2) Z(2)× Z(2) Z(4)

group G (2) F (4) E (6) E (7) E (8)

nG 14 52 78 133 248rank 2 4 6 7 8

center 1 1 Z(3) Z(2) 1

String tension from Polyakov loop correlators

0 rt = 0

t = β = 1/T

Φ(~x) = Tr P exp

(∫ β

0dt G0(~x , t)

〈Φ(0)∗Φ(r)〉 ∼ exp (−βV (r)) , V (r) ∼ σr

σ ≈ (0.4GeV)2 ≈ 105NAs strong as a cm-thick steel cable, but 13 orders of magnitude thinner.

0 rt = 0

t = β = 1/T

Φ(~x) = Tr P exp

(∫ β

0dt G0(~x , t)

〈Φ(0)∗Φ(r)〉 ∼ exp (−βV (r)) , V (r) ∼ σr

σ ≈ (0.4GeV)2 ≈ 105N

As strong as a cm-thick steel cable, but 13 orders of magnitude thinner.

0 rt = 0

t = β = 1/T

Φ(~x) = Tr P exp

(∫ β

0dt G0(~x , t)

〈Φ(0)∗Φ(r)〉 ∼ exp (−βV (r)) , V (r) ∼ σr

σ ≈ (0.4GeV)2 ≈ 105NAs strong as a cm-thick steel cable, but 13 orders of magnitude thinner.

How many people can be lifted by a Yang-Mills elevator?

@Gravity F = Mg

@ String tension σ

Mg = σ ≈ 105N, g ≈ 10m/sec2 ⇒ M ≈ 104kg ≈ 100 People

@Gravity F = Mg

@ String tension σ

@Gravity F = Mg

@ String tension σ

@Gravity F = Mg

@ String tension σ

Outline

Yang-Mills Theory

Conclusions

Low-energy effective string theory

The height variable ~h(x , t) points to the fluctuating string world-sheet in

the (d − 2) transverse dimensions (~h(0, t) = ~h(r , t) = 0).

S [~h] =

∫ β

2∂µ~h · ∂µ

~h, µ ∈ 0, 1,

V (r) = σr − π(d − 2)

24r+O(1/r3),

w2lo(r/2) =

d − 2

2πσlog(r/r0)

M. Luscher, K. Symanzik, P. Weisz, Nucl. Phys. B173 (1980) 365M. Luscher, Nucl. Phys. B180 (1981) 317M. Luscher, G. Munster, P. Weisz, Nucl. Phys. B180 (1981) 1

S [~h] =

∫ β

2∂µ~h · ∂µ

~h, µ ∈ 0, 1,

V (r) = σr − π(d − 2)

24r+O(1/r3),

w2lo(r/2) =

d − 2

2πσlog(r/r0)

S [~h] =

∫ β

2∂µ~h · ∂µ

~h, µ ∈ 0, 1,

V (r) = σr − π(d − 2)

24r+O(1/r3),

w2lo(r/2) =

d − 2

2πσlog(r/r0)

S [~h] =

∫ β

2∂µ~h · ∂µ

~h, µ ∈ 0, 1,

V (r) = σr − π(d − 2)

24r+O(1/r3),

w2lo(r/2) =

d − 2

2πσlog(r/r0)

S [~h] =

∫ β

2∂µ~h · ∂µ

~h, µ ∈ 0, 1,

V (r) = σr − π(d − 2)

24r+O(1/r3),

w2lo(r/2) =

d − 2

2πσlog(r/r0)

S [~h] =

∫ β

2∂µ~h · ∂µ

~h, µ ∈ 0, 1,

V (r) = σr − π(d − 2)

24r+O(1/r3),

w2lo(r/2) =

d − 2

2πσlog(r/r0)

S [~h] =

∫ β

2∂µ~h · ∂µ

~h, µ ∈ 0, 1,

V (r) = σr − π(d − 2)

24r+O(1/r3),

w2lo(r/2) =

d − 2

2πσlog(r/r0)

Effective string theory for d = 3 at the 2-loop level

S [h] =

∫ β

[∂µh∂µh −

8σ(∂µh∂µh)2

]Even to the next order, the action agrees with the one of theNambu-Goto string.O. Aharony and E. Karzbrun, JHEP 0906 (2009) 012

w2(r/2) = 〈h(r/2, t) h(r/2 + ε, t + ε′)〉

4πf (τ)

lo(r/2)− f (τ) + g(τ)

σ2r2,

f (τ) =E2(τ)− 4E2(2τ)

g(τ) = iπτ

(E2(τ)

12− q

)(f (τ) +

E2(τ)

E2(τ) = 1− 24∞∑

1− qn, q = exp(2πiτ), τ = iβ/2r

F. Gliozzi, M. Pepe, UJW, arXiv:1006.2252 [hep-lat]

Effective string theory for d = 3 at the 2-loop level

S [h] =

∫ β

[∂µh∂µh −

8σ(∂µh∂µh)2

]Even to the next order, the action agrees with the one of theNambu-Goto string.O. Aharony and E. Karzbrun, JHEP 0906 (2009) 012

w2(r/2) = 〈h(r/2, t) h(r/2 + ε, t + ε′)〉

4πf (τ)

lo(r/2)− f (τ) + g(τ)

σ2r2,

f (τ) =E2(τ)− 4E2(2τ)

g(τ) = iπτ

(E2(τ)

12− q

)(f (τ) +

E2(τ)

E2(τ) = 1− 24∞∑

1− qn, q = exp(2πiτ), τ = iβ/2r

F. Gliozzi, M. Pepe, UJW, arXiv:1006.2252 [hep-lat]

Outline

Yang-Mills Theory

Conclusions

Two-link operators

T (t)αβγδ = U0(0, t)∗αβU0(r , t)γδ

Multiplication law

T (t)T (t + a)αβγδ = T (t)αλγεT (t + a)λβεδ

Polyakov loop correlator

Φ(0)∗Φ(r) = T (0)T (a) . . .T (β − a)ααγγ

M. Luscher and P. Weisz, JHEP 0109 (2001) 010

Two-link operators

Multiplication law

Φ(0)∗Φ(r) = T (0)T (a) . . .T (β − a)ααγγ

Two-link operators

Multiplication law

Φ(0)∗Φ(r) = T (0)T (a) . . .T (β − a)ααγγ

Sub-lattice expectation value

[T (t) . . .T (t ′)] =1

∫DUsub T (t) . . .T (t ′) exp(−S [U]sub)

〈Φ(0)∗Φ(r)〉 = 〈[T (0)T (a)] . . . [T (β − 2a)T (β − a)]ααγγ〉

[T (t) . . .T (t ′)] =1

Outline

Yang-Mills Theory

Conclusions

Simulations in (2 + 1)-d SU(2) Yang-Mills theory

S [U] = − 1

∑x ,µ,ν

Tr[Uµ(x)Uν(x + µ)Uµ(x + ν)†Uν(x)†],

〈Φ(0)Φ(r)〉 =1

∫DU Φ(0)Φ(r) exp(−S [U]) ∼ exp(−βV (r))

2 4 6 8 10 12 14 16

V(R) = a + b R + c/Rspin 1/2

V (r) = σr − π24r + . . .

There is very good agreement with the effective string theory.The value of the string tension obtained on a 542 × 48 lattice at4/g2 = 9 is σ = 0.025897(15)/a2.

S [U] = − 1

∑x ,µ,ν

〈Φ(0)Φ(r)〉 =1

2 4 6 8 10 12 14 16

V(R) = a + b R + c/Rspin 1/2

V (r) = σr − π24r + . . .

S [U] = − 1

∑x ,µ,ν

〈Φ(0)Φ(r)〉 =1

2 4 6 8 10 12 14 16

V(R) = a + b R + c/Rspin 1/2

V (r) = σr − π24r + . . .

Computation of the string width

P(x) = Tr[U1(x)U0(x + 1)U1(x + 0)†U0(x)†],

C (x2) =〈Φ(0)Φ(r)P(x)〉〈Φ(0)Φ(r)〉

− 〈P(x)〉

0.88545

0.8855

0.88555

0.8856

0.88565

0.8857

0.88575

0.8858

0 2 4 6 8 10 12 14 16 18

fitdata

5 10 15 20 25

fit w2(r/2)

There is very good agreement with the effective string theory at thetwo-loop level. The values of the low-energy parameters areσ = 0.025897(15)/a2, r0 = 2.26(2)a = 0.364(4)/

√σ ≈ 0.2 fm.

F. Gliozzi, M. Pepe, UJW, Phys. Rev. Lett. 104 (2010) 232001

P(x) = Tr[U1(x)U0(x + 1)U1(x + 0)†U0(x)†],

C (x2) =〈Φ(0)Φ(r)P(x)〉〈Φ(0)Φ(r)〉

− 〈P(x)〉

0.88545

0.8855

0.88555

0.8856

0.88565

0.8857

0.88575

0.8858

0 2 4 6 8 10 12 14 16 18

fitdata

5 10 15 20 25

fit w2(r/2)

√σ ≈ 0.2 fm.

P(x) = Tr[U1(x)U0(x + 1)U1(x + 0)†U0(x)†],

C (x2) =〈Φ(0)Φ(r)P(x)〉〈Φ(0)Φ(r)〉

− 〈P(x)〉

0.88545

0.8855

0.88555

0.8856

0.88565

0.8857

0.88575

0.8858

0 2 4 6 8 10 12 14 16 18

fitdata

5 10 15 20 25

fit w2(r/2)

√σ ≈ 0.2 fm.

String width at finite temperature

w2(r/2) =1

2πσlog

2β+ . . .

At finite temperature, the effective string theory predicts a linear increaseof the width as one separates the static sources.

5 10 15 20 25

Again, there is excellent agreement with the effective string theory at thetwo-loop level, now without any adjustable parameters.F. Gliozzi, M. Pepe, UJW, to be published

w2(r/2) =1

2πσlog

2β+ . . .

5 10 15 20 25

w2(r/2) =1

2πσlog

2β+ . . .

5 10 15 20 25

Outline

Yang-Mills Theory

Conclusions

String breaking

and string decay (strand rupture)

Static potential between triplet (Q = 1) charges in (2+1)-d SU(2)Yang-Mills theory

2 4 6 8 10 12 14 16

V(R)spin 1

M. Pepe and UJW, Phys. Rev. Lett. 102 (2009) 191601

String breaking and string decay (strand rupture)

2 4 6 8 10 12 14 16

V(R)spin 1

2 4 6 8 10 12 14 16

V(R)spin 1

2 4 6 8 10 12 14 16

V(R)spin 1

String decay

2 4 6 8 10 12 14 16

fitQ = 1/2Q = 3/2

2 4 6 8 10 12 14

fitQ = 1/2Q = 3/2

Potential and force between quadruplet (Q = 3/2) charges

2 4 6 8 10 12 14 16

fitQ = 1Q = 2

2 4 6 8 10 12 14

fitQ = 1Q = 2

Potential and force between quintet (Q = 2) charges

String decay

2 4 6 8 10 12 14 16

fitQ = 1/2Q = 3/2

2 4 6 8 10 12 14

fitQ = 1/2Q = 3/2

Potential and force between quadruplet (Q = 3/2) charges

2 4 6 8 10 12 14 16

fitQ = 1Q = 2

2 4 6 8 10 12 14

fitQ = 1Q = 2

Potential and force between quintet (Q = 2) charges

Constituent gluon model: EQ,n(r) = σQr − cQ

r+ 2MQ,n

H1(r) =

(E1,0(r) A

A E0,1(r)

H3/2(r) =

(E3/2,0(r) B

B E1/2,1(r)

H2(r) =

E2,0(r) C 0C E1,1(r) A0 A E0,2(r)

Q σQa2 σQ/σ 4Q(Q + 1)/3

1/2 0.06397(3) 1 1

1 0.144(1) 2.25(2) 8/3

3/2 0.241(5) 3.77(8) 5

2 0.385(5) 6.02(8) 8

We observe deviations from Casimir scaling.

Constituent gluon model: EQ,n(r) = σQr − cQ

r+ 2MQ,n

H1(r) =

(E1,0(r) A

A E0,1(r)

H3/2(r) =

(E3/2,0(r) B

B E1/2,1(r)

H2(r) =

E2,0(r) C 0C E1,1(r) A0 A E0,2(r)

Q σQa2 σQ/σ 4Q(Q + 1)/3

1/2 0.06397(3) 1 1

1 0.144(1) 2.25(2) 8/3

3/2 0.241(5) 3.77(8) 5

2 0.385(5) 6.02(8) 8

We observe deviations from Casimir scaling.

Masses and mass differences: ∆Q,n = MQ−1,n+1 −MQ,n

Q MQ,0a MQ−1,1a MQ−2,2a ∆Q,0a ∆Q−1,1a

1/2 0.109(1) — — — —

1 0.37(3) 1.038(1) — 0.67(3) —

3/2 0.72(5) 1.32(5) — 0.60(5) —

2 1.04(3) 1.71(3) 2.42(1) 0.67(3) 0.71(3)

Constituent gluon mass:MG = 0.65(4)/a = 2.6(2)

√σ ≈ 1 GeV

0+ glueball mass: M0+ = 1.198(25)/aH. Meyer and M. Teper, Nucl. Phys. B668 (2003) 111

A simple constituent gluon model describes the data rather well.

Q MQ,0a MQ−1,1a MQ−2,2a ∆Q,0a ∆Q−1,1a

1/2 0.109(1) — — — —

1 0.37(3) 1.038(1) — 0.67(3) —

3/2 0.72(5) 1.32(5) — 0.60(5) —

2 1.04(3) 1.71(3) 2.42(1) 0.67(3) 0.71(3)

√σ ≈ 1 GeV

Q MQ,0a MQ−1,1a MQ−2,2a ∆Q,0a ∆Q−1,1a

1/2 0.109(1) — — — —

1 0.37(3) 1.038(1) — 0.67(3) —

3/2 0.72(5) 1.32(5) — 0.60(5) —

2 1.04(3) 1.71(3) 2.42(1) 0.67(3) 0.71(3)

√σ ≈ 1 GeV

Q MQ,0a MQ−1,1a MQ−2,2a ∆Q,0a ∆Q−1,1a

1/2 0.109(1) — — — —

1 0.37(3) 1.038(1) — 0.67(3) —

3/2 0.72(5) 1.32(5) — 0.60(5) —

2 1.04(3) 1.71(3) 2.42(1) 0.67(3) 0.71(3)

√σ ≈ 1 GeV

Outline

Yang-Mills Theory

Conclusions

Exceptional Lie Group G (2) ⊂ SO(7)

ΩabΩac = δbc , Tabc = Tdef ΩdaΩebΩfc ,

T127 = T154 = T163 = T235 = T264 = T374 = T576 = 1

−1/2 3

−1/2 1/2

−1/ 3

−−

1/2 1−1 −1/2

−1/ 2 3

− 3 / 2

7 = 3 ⊕ 3 ⊕ 1,

14 = 8 ⊕ 3 ⊕ 3

The fact that G (2) has a trivial center causes exceptional confinement.K. Holland, P. Minkowski, M. Pepe, UJW, Nucl. Phys. B668 (2003) 207

T127 = T154 = T163 = T235 = T264 = T374 = T576 = 1

−1/2 3

−1/2 1/2

−1/ 3

−−

1/2 1−1 −1/2

−1/ 2 3

− 3 / 2

7 = 3 ⊕ 3 ⊕ 1,

14 = 8 ⊕ 3 ⊕ 3

T127 = T154 = T163 = T235 = T264 = T374 = T576 = 1

−1/2 3

−1/2 1/2

−1/ 3

−−

1/2 1−1 −1/2

−1/ 2 3

− 3 / 2

7 = 3 ⊕ 3 ⊕ 1, 14 = 8 ⊕ 3 ⊕ 3

T127 = T154 = T163 = T235 = T264 = T374 = T576 = 1

−1/2 3

−1/2 1/2

−1/ 3

−−

1/2 1−1 −1/2

−1/ 2 3

− 3 / 2

7 = 3 ⊕ 3 ⊕ 1, 14 = 8 ⊕ 3 ⊕ 3

Casimir Scaling in G (2) Yang-Mills Theory

14 ⊗ 14 ⊗ 14 = 1 ⊕ 7 ⊕ 5 14 ⊕ 3 27 ⊕ 2 64⊕ 4 77 ⊕ 3 77′ ⊕ 182 ⊕ 3 189⊕ 273 ⊕ 2 448

Three gluons can screen a single quark.

0 1 2 3 4 5 6 7

71427647777’

L. Liptak and S. Olejnik, Phys. Rev. D78 (2008) 074501B. H. Wellegehausen, A. Wipf, and C. Wozar, arXiv:1006.2305

14 ⊗ 14 ⊗ 14 = 1 ⊕ 7 ⊕ 5 14 ⊕ 3 27 ⊕ 2 64⊕ 4 77 ⊕ 3 77′ ⊕ 182 ⊕ 3 189⊕ 273 ⊕ 2 448

0 1 2 3 4 5 6 7

71427647777’

14 ⊗ 14 ⊗ 14 = 1 ⊕ 7 ⊕ 5 14 ⊕ 3 27 ⊕ 2 64⊕ 4 77 ⊕ 3 77′ ⊕ 182 ⊕ 3 189⊕ 273 ⊕ 2 448

0 1 2 3 4 5 6 7

71427647777’

String Breaking in 3-d G (2) Yang-Mills Theory

14 ⊗ 14 ⊗ 14 = 1 ⊕ 7 ⊕ 5 14 ⊕ 3 27 ⊕ 2 64⊕ 4 77 ⊕ 3 77′ ⊕ 182 ⊕ 3 189⊕ 273 ⊕ 2 448

One or two gluons cannot screen a fundamental quark to a smallercharge, but three can.

B. H. Wellegehausen, A. Wipf, and C. Wozar, arXiv:1006.2305The fundamental string breaks by the simultaneous creation of six gluons,without intermediate string decay.

14 ⊗ 14 ⊗ 14 = 1 ⊕ 7 ⊕ 5 14 ⊕ 3 27 ⊕ 2 64⊕ 4 77 ⊕ 3 77′ ⊕ 182 ⊕ 3 189⊕ 273 ⊕ 2 448

B. H. Wellegehausen, A. Wipf, and C. Wozar, arXiv:1006.2305

The fundamental string breaks by the simultaneous creation of six gluons,without intermediate string decay.

14 ⊗ 14 ⊗ 14 = 1 ⊕ 7 ⊕ 5 14 ⊕ 3 27 ⊕ 2 64⊕ 4 77 ⊕ 3 77′ ⊕ 182 ⊕ 3 189⊕ 273 ⊕ 2 448

Outline

Yang-Mills Theory

Conclusions

• In Yang-Mills theory there is a systematic low-energy effective stringtheory (analogous to chiral perturbation theory in QCD), which describesthe dynamics of the Goldstone modes of the spontaneously brokentranslation symmetry of the string world-sheet.

• The effective theory has been tested at the two-loop level with veryhigh precision using the Luscher and Weisz multi-level simulationtechnique, which led to the determination of low-energy parameter r0.

• Strings connecting static charges in higher representations may decaybefore they break completely.

• A constitutent gluon model accounts for the “brown muck”surrounding a screened color charge.

• The exceptional group G (2) is the smallest one with a trivial center.One observes Casimir scaling at the few percent accuracy level.A fundamental 7 quark can bescreened by three adjoint 14 gluons.

The fundamental G (2) string breaks by the simultaneous creation of six

gluons without string decay.

Conclusions

Yang-Mills elevator as a metaphor for our workshop

@Gravity F = Mg

@ String tension σ

Thanks for organizing a very nice workshop thatcontributes to quickly elevate us to a higher level ofunderstanding of strongly interacting field theories!

@Gravity F = Mg

@ String tension σ

@Gravity F = Mg

@ String tension σ

@Gravity F = Mg

@ String tension σ

@Gravity F = Mg

@ String tension σ

@Gravity F = Mg

@ String tension σ

@Gravity F = Mg

@ String tension σ

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